Abstract
An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G. The oriented diameter of a graph G is the smallest diameter among all the orientations of G. The maximum oriented diameter of a family of graphs \(\mathscr {F}\) is the maximum oriented diameter among all the graphs in \(\mathscr {F}\). Chvátal and Thomassen [JCTB, 1978] gave a lower bound of \(\frac{1}{2}{d^2+d}\) and an upper bound of \(2d^2+2d\) for the maximum oriented diameter of the family of 2-edge connected graphs of diameter d. We improve this upper bound to \( 1.373 d^2 + 6.971d-1 \), which outperforms the former upper bound for all values of d greater than or equal to 8. For the family of 2-edge connected graphs of diameter 3, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of 9 and 11 respectively. For the family of 2-edge connected graphs of diameter 4, the bounds provided by Chvátal and Thomassen are 12 and 40 and no better bounds were known. By extending the method we used for diameter d graphs, along with an asymmetric extension of a technique used by Chvátal and Thomassen, we have improved this upper bound to 21.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Robbins, H.E.: A theorem on graphs, with an application to a problem of traffic control. Am. Math. Monthly 46(5), 281–283 (1939)
Chvátal, V., Thomassen, C.: Distances in orientations of graphs. J. Comb. Theory Ser. B 24(1), 61–75 (1978)
Chung, F.R.K., Garey, M.R., Tarjan, R.E.: Strongly connected orientations of mixed multigraphs. Networks 15(4), 477–484 (1985)
Fomin, F.V., Matamala, M., Prisner, E., Rapaport, I.: AT-free graphs: linear bounds for the oriented diameter. Discrete Appl. Math. 141(1–3), 135–148 (2004)
Fomin, F.V., Matamala, M., Rapaport, I.: Complexity of approximating the oriented diameter of chordal graphs. J. Graph Theory 45(4), 255–269 (2004)
Eggemann, N., Noble, S.D.: Minimizing the oriented diameter of a planar graph. Electron. Notes Discrete Math. 34, 267–271 (2009)
Bau, S., Dankelmann, P.: Diameter of orientations of graphs with given minimum degree. Eur. J. Comb. 49, 126–133 (2015)
Surmacs, M.: Improved bound on the oriented diameter of graphs with given minimum degree. Eur. J. Comb. 59, 187–191 (2017)
Kwok, P.K., Liu, Q., West, D.B.: Oriented diameter of graphs with diameter 3. J. Comb. Theory Ser. B 100(3), 265–274 (2010)
Huang, X., Li, H., Li, X., Sun, Y.: Oriented diameter and rainbow connection number of a graph. Discret. Math. Theor. Comput. Sci. 16(3), 51–60 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Babu, J., Benson, D., Rajendraprasad, D., Vaka, S.N. (2020). An Improvement to Chvátal and Thomassen’s Upper Bound for Oriented Diameter. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-50026-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-50025-2
Online ISBN: 978-3-030-50026-9
eBook Packages: Computer ScienceComputer Science (R0)